Ok, so here's the issue. I've got these two alternating series and the problem states that one is conditional and one is absolutely convergent. The problem I'm having is that, using both the root test and the ratio test, I am coming up with both as absolutely convergent. So I must be doing something wrong. Summary of my work below:
Alternating Series (1/2)^n
Converges by AST: decreasing, positive, limit of zero.
Ratio test = 1/2 < 1 so absolute convergence.
Root test = 1/2 < 1 so absolute convergence
Alternating series 1/sqrt(n)
Converges by AST: decreasing, positive, limit of zero.
Ratio test = 1 so no conclusion can be drawn
Root test = lim of the n root of 1/sqrt(n) <1 so absolutely convergent
There's a mistake somewhere but I'm clearly missing it. Thanks for any help!
Best Answer
The mistake is that in fact $$\lim_{n\to\infty} (\sqrt{n})^{1/n} = 1,$$ so the root test is also inconclusive. A simple comparison with for example the harmonic series shows that the second is not absolutely convergent.